Stay ahead of learning milestones! Enroll in a class over the summer!

G
Topic
First Poster
Last Poster
k a May Highlights and 2025 AoPS Online Class Information
jlacosta   0
May 1, 2025
May is an exciting month! National MATHCOUNTS is the second week of May in Washington D.C. and our Founder, Richard Rusczyk will be presenting a seminar, Preparing Strong Math Students for College and Careers, on May 11th.

Are you interested in working towards MATHCOUNTS and don’t know where to start? We have you covered! If you have taken Prealgebra, then you are ready for MATHCOUNTS/AMC 8 Basics. Already aiming for State or National MATHCOUNTS and harder AMC 8 problems? Then our MATHCOUNTS/AMC 8 Advanced course is for you.

Summer camps are starting next month at the Virtual Campus in math and language arts that are 2 - to 4 - weeks in duration. Spaces are still available - don’t miss your chance to have an enriching summer experience. There are middle and high school competition math camps as well as Math Beasts camps that review key topics coupled with fun explorations covering areas such as graph theory (Math Beasts Camp 6), cryptography (Math Beasts Camp 7-8), and topology (Math Beasts Camp 8-9)!

Be sure to mark your calendars for the following upcoming events:
[list][*]May 9th, 4:30pm PT/7:30pm ET, Casework 2: Overwhelming Evidence — A Text Adventure, a game where participants will work together to navigate the map, solve puzzles, and win! All are welcome.
[*]May 19th, 4:30pm PT/7:30pm ET, What's Next After Beast Academy?, designed for students finishing Beast Academy and ready for Prealgebra 1.
[*]May 20th, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 1 Math Jam, Problems 1 to 4, join the Canada/USA Mathcamp staff for this exciting Math Jam, where they discuss solutions to Problems 1 to 4 of the 2025 Mathcamp Qualifying Quiz!
[*]May 21st, 4:00pm PT/7:00pm ET, Mathcamp 2025 Qualifying Quiz Part 2 Math Jam, Problems 5 and 6, Canada/USA Mathcamp staff will discuss solutions to Problems 5 and 6 of the 2025 Mathcamp Qualifying Quiz![/list]
Our full course list for upcoming classes is below:
All classes run 7:30pm-8:45pm ET/4:30pm - 5:45pm PT unless otherwise noted.

Introductory: Grades 5-10

Prealgebra 1 Self-Paced

Prealgebra 1
Tuesday, May 13 - Aug 26
Thursday, May 29 - Sep 11
Sunday, Jun 15 - Oct 12
Monday, Jun 30 - Oct 20
Wednesday, Jul 16 - Oct 29

Prealgebra 2 Self-Paced

Prealgebra 2
Wednesday, May 7 - Aug 20
Monday, Jun 2 - Sep 22
Sunday, Jun 29 - Oct 26
Friday, Jul 25 - Nov 21

Introduction to Algebra A Self-Paced

Introduction to Algebra A
Sunday, May 11 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Wednesday, May 14 - Aug 27
Friday, May 30 - Sep 26
Monday, Jun 2 - Sep 22
Sunday, Jun 15 - Oct 12
Thursday, Jun 26 - Oct 9
Tuesday, Jul 15 - Oct 28

Introduction to Counting & Probability Self-Paced

Introduction to Counting & Probability
Thursday, May 15 - Jul 31
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Wednesday, Jul 9 - Sep 24
Sunday, Jul 27 - Oct 19

Introduction to Number Theory
Friday, May 9 - Aug 1
Wednesday, May 21 - Aug 6
Monday, Jun 9 - Aug 25
Sunday, Jun 15 - Sep 14
Tuesday, Jul 15 - Sep 30

Introduction to Algebra B Self-Paced

Introduction to Algebra B
Tuesday, May 6 - Aug 19
Wednesday, Jun 4 - Sep 17
Sunday, Jun 22 - Oct 19
Friday, Jul 18 - Nov 14

Introduction to Geometry
Sunday, May 11 - Nov 9
Tuesday, May 20 - Oct 28
Monday, Jun 16 - Dec 8
Friday, Jun 20 - Jan 9
Sunday, Jun 29 - Jan 11
Monday, Jul 14 - Jan 19

Paradoxes and Infinity
Mon, Tue, Wed, & Thurs, Jul 14 - Jul 16 (meets every day of the week!)

Intermediate: Grades 8-12

Intermediate Algebra
Sunday, Jun 1 - Nov 23
Tuesday, Jun 10 - Nov 18
Wednesday, Jun 25 - Dec 10
Sunday, Jul 13 - Jan 18
Thursday, Jul 24 - Jan 22

Intermediate Counting & Probability
Wednesday, May 21 - Sep 17
Sunday, Jun 22 - Nov 2

Intermediate Number Theory
Sunday, Jun 1 - Aug 24
Wednesday, Jun 18 - Sep 3

Precalculus
Friday, May 16 - Oct 24
Sunday, Jun 1 - Nov 9
Monday, Jun 30 - Dec 8

Advanced: Grades 9-12

Olympiad Geometry
Tuesday, Jun 10 - Aug 26

Calculus
Tuesday, May 27 - Nov 11
Wednesday, Jun 25 - Dec 17

Group Theory
Thursday, Jun 12 - Sep 11

Contest Preparation: Grades 6-12

MATHCOUNTS/AMC 8 Basics
Friday, May 23 - Aug 15
Monday, Jun 2 - Aug 18
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

MATHCOUNTS/AMC 8 Advanced
Sunday, May 11 - Aug 10
Tuesday, May 27 - Aug 12
Wednesday, Jun 11 - Aug 27
Sunday, Jun 22 - Sep 21
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Problem Series
Friday, May 9 - Aug 1
Sunday, Jun 1 - Aug 24
Thursday, Jun 12 - Aug 28
Tuesday, Jun 17 - Sep 2
Sunday, Jun 22 - Sep 21 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Monday, Jun 23 - Sep 15
Tues & Thurs, Jul 8 - Aug 14 (meets twice a week!)

AMC 10 Final Fives
Sunday, May 11 - Jun 8
Tuesday, May 27 - Jun 17
Monday, Jun 30 - Jul 21

AMC 12 Problem Series
Tuesday, May 27 - Aug 12
Thursday, Jun 12 - Aug 28
Sunday, Jun 22 - Sep 21
Wednesday, Aug 6 - Oct 22

AMC 12 Final Fives
Sunday, May 18 - Jun 15

AIME Problem Series A
Thursday, May 22 - Jul 31

AIME Problem Series B
Sunday, Jun 22 - Sep 21

F=ma Problem Series
Wednesday, Jun 11 - Aug 27

WOOT Programs
Visit the pages linked for full schedule details for each of these programs!


MathWOOT Level 1
MathWOOT Level 2
ChemWOOT
CodeWOOT
PhysicsWOOT

Programming

Introduction to Programming with Python
Thursday, May 22 - Aug 7
Sunday, Jun 15 - Sep 14 (1:00 - 2:30 pm ET/10:00 - 11:30 am PT)
Tuesday, Jun 17 - Sep 2
Monday, Jun 30 - Sep 22

Intermediate Programming with Python
Sunday, Jun 1 - Aug 24
Monday, Jun 30 - Sep 22

USACO Bronze Problem Series
Tuesday, May 13 - Jul 29
Sunday, Jun 22 - Sep 1

Physics

Introduction to Physics
Wednesday, May 21 - Aug 6
Sunday, Jun 15 - Sep 14
Monday, Jun 23 - Sep 15

Physics 1: Mechanics
Thursday, May 22 - Oct 30
Monday, Jun 23 - Dec 15

Relativity
Mon, Tue, Wed & Thurs, Jun 23 - Jun 26 (meets every day of the week!)
0 replies
jlacosta
May 1, 2025
0 replies
IMO Genre Predictions
ohiorizzler1434   48
N a few seconds ago by jkim0656
Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
48 replies
ohiorizzler1434
May 3, 2025
jkim0656
a few seconds ago
Concurrency from isogonal Mittenpunkt configuration
MarkBcc168   17
N an hour ago by Ilikeminecraft
Source: Fake USAMO 2020 P3
Let $\triangle ABC$ be a scalene triangle with circumcenter $O$, incenter $I$, and incircle $\omega$. Let $\omega$ touch the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ at points $D$, $E$, and $F$ respectively. Let $T$ be the projection of $D$ to $\overline{EF}$. The line $AT$ intersects the circumcircle of $\triangle ABC$ again at point $X\ne A$. The circumcircles of $\triangle AEX$ and $\triangle AFX$ intersect $\omega$ again at points $P\ne E$ and $Q\ne F$ respectively. Prove that the lines $EQ$, $FP$, and $OI$ are concurrent.

Proposed by MarkBcc168.
17 replies
MarkBcc168
Apr 28, 2020
Ilikeminecraft
an hour ago
Inequality with a,b,c
GeoMorocco   7
N an hour ago by lele0305
Source: Morocco Training
Let $   a,b,c   $ be positive real numbers such that : $   ab+bc+ca=3   $ . Prove that : $$\frac{\sqrt{1+a^2}}{1+ab}+\frac{\sqrt{1+b^2}}{1+bc}+\frac{\sqrt{1+c^2}}{1+ca}\ge \sqrt{\frac{3(a+b+c)}{2}}$$
7 replies
GeoMorocco
Apr 11, 2025
lele0305
an hour ago
Property of the divisors of k^3 - 2
Scilyse   2
N 2 hours ago by Assassino9931
Source: KoMaL A. 892
Given two integers, $k$ and $d$ such that $d$ divides $k^3 - 2$. Show that there exists integers $a$, $b$, $c$ satisfying $d = a^3 + 2b^3 + 4c^3 - 6abc$.

Proposed by Csongor Beke and László Bence Simon, Cambridge
2 replies
Scilyse
Jan 13, 2025
Assassino9931
2 hours ago
No more topics!
Parallelity and equal angles given, wanted an angle equality
BarisKoyuncu   5
N Apr 24, 2025 by SleepyGirraffe
Source: 2022 Turkey JBMO TST P4
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
5 replies
BarisKoyuncu
Mar 15, 2022
SleepyGirraffe
Apr 24, 2025
Parallelity and equal angles given, wanted an angle equality
G H J
G H BBookmark kLocked kLocked NReply
Source: 2022 Turkey JBMO TST P4
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BarisKoyuncu
577 posts
#1 • 4 Y
Y by teomihai, HWenslawski, son7, oralayhan
Given a convex quadrilateral $ABCD$ such that $m(\widehat{ABC})=m(\widehat{BCD})$. The lines $AD$ and $BC$ intersect at a point $P$ and the line passing through $P$ which is parallel to $AB$, intersects $BD$ at $T$. Prove that
$$m(\widehat{ACB})=m(\widehat{PCT})$$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
BarisKoyuncu
577 posts
#2 • 3 Y
Y by teomihai, HWenslawski, oralayhan
Let the line passing through $D$ and parallel to $AB$ intersects $BP$ and $AT$ at $K$ and $L$, respectively. We have
$$\frac{|DK|}{|AB|}=\frac{|PD|}{|PA|}=\frac{|PD|}{|PA|}\cdot\frac{|PT|}{|DL|}\cdot\frac{|DL|}{|PT|}=\frac{|PD|}{|PA|}\cdot\frac{|AP|}{|AD|}\cdot\frac{|DL|}{|PT|}=$$$$\frac{|PD|}{|AD|}\cdot\frac{|DL|}{|PT|}=\frac{|PT|}{|AB|}\cdot\frac{|DL|}{|PT|}=\frac{|DL|}{|AB|}\Rightarrow |DK|=|DL|$$Also, $\angle DCK=\angle ABC=\angle DKC\Rightarrow |DK|=|DC|$.
Therefore, $\angle LCK=90^{\circ}$.
Hence, it suffices to prove that the angle bisector of $\angle ACT$ is $CL$.
Let $TA\cap PB=M$. We have
$$\frac{|MA|}{|MT|}=\frac{|AB|}{|TP|}=\frac{|AD|}{|DP|}=\frac{|LA|}{|LT|}\Rightarrow (M,L;A,T)=-1$$Also, $\angle LCM=90^{\circ}$. Hence, we conclude that the angle bisector of $\angle ACT$ is $CL$.
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
hakN
429 posts
#3 • 2 Y
Y by HWenslawski, oralayhan
Let $AB \cap CD = Q$ and $TC \cap AB = E$. Note that by the given angle condition, $\triangle QBC$ is isosceles with $QB = QC$. By angle chasing, it suffices to prove $\angle QCA = \angle CEQ$, or that $QC^2 = QA \cdot QE$.

By Menelaus and similarity of triangles $\triangle BCE \sim \triangle PCT$ and $\triangle ADB \sim \triangle PDT$, we have $$1 = \dfrac{QA}{QB} \cdot \dfrac{BC}{CP} \cdot \dfrac{PD}{DA} = \dfrac{QA}{QB} \cdot \dfrac{BE}{PT} \cdot \dfrac{PT}{AB} \implies QA \cdot BE = QB \cdot AB.$$
Using the fact that $BE = QE - QB$, this gives $QA \cdot QE = QB^2 = QC^2$, which implies the result. $\square$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
bin_sherlo
719 posts
#4
Y by
Let $TA\cap BC=G, TA\cap CF=K$, foot of the altitude from $A$ to $BC$ be $H$, the perpendicular at $C$ to $BC$ meet $BA$ at $F$, foot of the altitude from $T$ to $BC$ be $L$, $BA\cap CD=E, TC\cap AB=R$, $GD$ meet $BF, TB$ at $M,N$ respectively.
We want to show that $BC$ is the angle bisector of $\angle RCA \iff (R,A;B,F)=-1$
\[(R,A;B,F)=(TC,TA;TB,TF)=(C,G;B,TF\cap BC)=(FC,FG;FB,FT)=(K,G;A,T)=(C,G;H,L)\]\[\frac{GH}{GL}=\frac{AH}{TL}=\frac{AB}{TP}=\frac{GB}{GP}=\frac{GM}{GN}=\frac{DM}{DN}=\frac{CH}{CL}\]As desired.$\blacksquare$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
bin_sherlo
719 posts
#5
Y by
$AB\cap CD=Q$
We will use method of moving points.
Take $QBCD$ fixed. Animate $A$ over $QB$. Denote $R(XY,ZT)$ as reflection of $XY$ to $ZT$.
\[f:A\rightarrow AC\rightarrow R(AC,BC)\rightarrow R(AC,BC)\cap BQ\]\[g: A\rightarrow AD\rightarrow AD\cap BC=T\rightarrow T(QB)_{\infty}\cap BD=P\rightarrow PC\cap QB\]$f,g$ has degree $2$.
$i)A=B$
$f: B\rightarrow BC\rightarrow BC\rightarrow B$
$g: B\rightarrow BD\rightarrow B\rightarrow B\rightarrow B$
So they are same at $A=B$.

$ii)A=Q$
Let the parallel line from $C$ to $QB$ intersect $BD$ at $S$.
$f:Q\rightarrow QC\rightarrow Q'C\rightarrow QB_{\infty}$
$g: Q\rightarrow QD\rightarrow C\rightarrow C(QB)_{\infty}\cap BD=S\rightarrow SC\cap QB=QB_{\infty}$
So they are same at $A=Q$.

Let the parallel from $D$ to $BC$ intersect $QB$ at $E$ and the parallel from $C$ to $BD$ intersect $QB$ at $F$.
$iii)A=E$
$f: E\rightarrow EC\rightarrow FC\rightarrow F$
$g: E\rightarrow ED\rightarrow BC_{\infty}\rightarrow (BC)_{\infty}(QB)_{\infty}\cap BD=BD_{\infty}\rightarrow F$
Thus $f,g$ are same at $3$ points as desired.$\blacksquare$
Z K Y
The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
SleepyGirraffe
1 post
#6
Y by
Let (ABC) intersect (CPT) in point E.
Claim: T,A,E are colinear!
Proof:
We have that : angle ABC = 180- angle TPC = angle TEC (1)
Also, CAEB is cyclic so angle ABC = angle AEC (2)
From (1) and (2) we get that :
angle TEC = angle AEC wich concludes our proof
Now, using this claim the problem is basically solved with a simple angle-chasing:
I will leave it up to the reader to figure it out
Z K Y
N Quick Reply
G
H
=
a